AN INTRODUCTION TO EXCEL



An Introduction to Excel

Calculate and Plot Radiation Spectra Emitted by a Blackbody.

This worksheet leads you through an excel environment to solve a problem. In this case, the problem calculates and plots the energy intensity emitted by a blackbody, as well as calculates the total energy and wavelength for the maximum energy based on Planck Radiation, Wien Displacement, and Stefan-Boltzmann Laws.

These directions assume that you have some familiarity with Microsoft Excel. If you do, then you should be able to follow these directions with very little trouble. If you are completely unfamiliar with Excel, try the Excel Tutorial first at:



If you are familiar with Excel, then try the assignment.

Assignment:

1. Plot the energy intensity spectrums on the same scatter graph that are emitted by blackbodies at three different surface temperatures using the Planck Radiation Law:

2. Also, calculate the total energy emitted (the area under the energy intensity curve) and the wavelength at the peak of the energy intensity distribution using the Wien Distribution and Stefan-Boltzmann Laws. The three temperatures under consideration are:

a) 6,000 ºK

b) 1,000 ºK, and

c) 300 ºK

Background:

The mean properties of electromagnetic radiation emitted from matter are approximated by a simple set of radiation laws, and apply when the emitter is a blackbody radiator. Although stars, humans and the earth are not perfect blackbodies, these laws do a reasonably good approximation of the radiation emitted by all three objects.

Planck Radiation Law:

This law describes the intensity of the radiation per unit surface area emitted by a black body as a function of wavelength (cm) and surface temperature (ºK).

Where, E(λ,T) is the energy intensity emitted at wavelength λ (cm) and temperature T (ºKelvin).

h 6.625 x 10-27 ergs-sec Plancks Constant

k 1.38 x 10-16 ergs/K Boltzmann Constant

c 3.00 x 1010 cm.sec Speed of Light

e 2.71828… the base for the natural logarithm (next to ln x on most calculators).

Two additional Laws are critical for this assignment, Wien Displacement and Stefan-Boltzmann Laws. They are:

The Stefan-Boltzmann Law:

Where E (in ergs-cm2/sec) is the total energy by all wavelengths emitted from the blackbody at temperature T (in ºKelvin).

σ 5.5705 x 10-5 erg-cm2-K-4-sec-1 Stefan-Blotzmann Constant

Wien Displacement Law:

Where λmax is the wavelength (in microns, μm) of the peak radiation emitted by a blackbody at temperature T (in ºKelvin).

Step 1. Start Excel.

Select Microsoft Excel from the start menu or double click on the Microsoft Excel icon. Any empty worksheet should appear on your screen. An example from Excel-2007 is shown below. Your menu bar might look a bit different, because you might have different options set up and/or your version of Excel might be different.

Step 2. Name, Date & Assignment.

Lets enter some labels in cells A2-A4.

In A2 enter Name:; A3 enter Date:; and A4 enter Assignment:

In B2-B4, type in your name, the date and the assignment corresponding to the labels.

Finally you might want to save this file. Choose a convenient location on your M: Drive, a location accessible by any computer logged onto the campus network. When prompted, choose a file name that will trigger your memory many months/years into the future, for example, choose Env-170 PSet-1 Problem 2.

You may need to change column widths (double click on the line between the A and B columns to automatically adjust the column widths to the entered information. You could also change the cell alignment by selecting the column, and clicking on the alignment icons in the Home menu. In my example, these entries were formatted in Bold.

Your worksheet should look like the reproduction below.

Step 3. Constants, Headings for Total Energy & λmax Calculations, Wavelengths & Energy Intensities, and Enter the Temperatures.

In this problem, you need Planck’s Constant, h; Boltzmann Constant, k; the speed of light, c; and the Stefan-Boltzmann Constant, σ. Lets enter these labels in column A, and their values in column B. Please note that Excel does not understand “x 10” in scientific notation. Instead use a capital “E” in its place. Therefore, 1.38 x 10-16 is entered as 1.38E-16.

Below these constants in row 11 & 12, lets add labels for the Total Energy and λmax calculations in column B.

Below the labels, enter additional labels for five successive columns of calculated information in row 14, columns A-E, Wavelength (cm), λ (μm), Energy, Energy, Energy. We have two columns of wavelengths so that we remember to use wavelengths in centimeters in the formulas but plot wavelengths in microns (μm). The three columns of Energy provide space for the

Energy Intensity values calculated for the three different objects.

Finally, enter the object surface temperatures, 6,000, 1,000 and 300 in cells C15, D15, and E15, respectively, below the energy labels.

Your worksheet should look like the reproduced section below:

Step 4. Formulas to Calculate Total Energy & λmax at Each Temperature.

Formulas in Excel start with an “=” equal sign. Enter and “=” equal sign followed by the Stefan-Boltzmann Law formula to calculate the Total Energy in cell C11. Below, the equation is shown in Excel format as well as the corresponding terms in and formula.

= $B9 * C15 ^ 4

You can either type the cell reference $B9 for the Stefan-Boltzmann Constant, or just click on the cell containing the value of this constant (and add the $ later). When entered, the answer to the formula is shown in the cell (7.35 x 1010), but the formulas still remains visible in the formula bar.

Drag the formula to the two cells towards the right to calculate the Total Energy for the other two temperatures. Do this by moving the mouse over the lower right corner of the cell, the cursor should change to a small plus sign, then click the left mouse button and drag the formula two cells to the right.

Note that the dollar sign in $B9 distinguishes this reference as an absolute reference, instead of a relative reference. Absolute references do not change as the formula is copied and pasted to other cells, in this case, two cells towards the right. Relative references do change. Therefore, each calculation uses the same Stefan-Boltzmann constant, but different temperatures. The formula in:

D11 should be = $B9 * D15^4, and (where “*” is the multiplication sign in Excel)

E11 should be = $B9 * E15^4. (and “^” is the power sign in Excel)

Now, calculate the λmax values for each temperature. In cell C12, enter the Wien’s Law formula. Remember start the formula with an “=” equal sign followed by the formula. Below, the equation is shown in Excel format as well as the corresponding terms in and formula.

= 2897 / C15

Drag the formula to the right two cells. The C15 should change to D15 and E15 in the subsequent cells and will calculate the wavelengths for the other two temperatures.

Your worksheet should look like the reproduced section to the left:

How do λmax and Energies change as temperatures decrease?

Step 5: Enter the Wavelengths and Calculate the Energy Intensities.

Enter the wavelengths from 0.05 μm to 15 microns in 0.05 μm increments. To do this, enter 0.05 in cell B16, enter 0.1 in cell B17, highlight cells B16 & B17, drag the sequence down until you get to 15 μm (B315). Excel will maintain the step as you drag the numbers downward.

Calculate the wavelengths in centimeters (cm), knowing that 1 cm = 10,000 μm. In cell A16, enter: = B16/10000. You can click on the B16 cell if you want, then / then 10000. The answer is 5.0 x 10-6.

Drag this formula down, so each wavelength in μm has a corresponding wavelength in cm.

Enter Planck’s Law formula into cell C16 to calculate the Energy Intensity for 0.05 μm and 6,000 ºK. The formula is:

= 2 * $B$6 * $B$8^2 / $A16^5 / (EXP($B$6 * $B$8/$A16/$B$7/C$15) – 1)

The various dollar signs fix either the row or column or both as the formula is dragged, so that the formula can be dragged downward and to the right and still use the correct constants, wavelengths and temperatures. Also notice that an extra “/” is in the exponent for “e”, and “e” is EXP() in Excel. The answer is 5.36 x 100, in normal notation: 5.36.

Use scientific notation, because some numbers get really really large and others really really small. If your answer is correct, then drag this formula both downward and towards the right, to calculate the energy intensity for every wavelength and every temperature, respectively.

Your worksheet should look like the reproduced section to the left:

Notice the Energy Intensity for 0.05 μm and 300 ºK is #NUM!

This is an Excel error message indicating that the number you tried to calculate is way too small for Excel to calculate. Replace the #NUM! with 1.0E-300

Once replaced, this cell should show 1.00E-300 (a very small number).

Step 6: Graph the Energy Intensities vs. Wavelength

Highlight the wavelengths and energy intensities to graph them, starting with B17 and ending with E315. The highlight should include the row temperatures so that they will be placed in the legend.

Choose Insert>Scatter Chart with Smoot Lines. The graph should appear like the example below:

It looks like we need to fix the y-axis and x-axis scales to see all of the plots, and we need labels.

To change the x-axis scale, right-click on the x-axis, and select the format axis option. Under Axis Options, change the Maximum to 15 and Major Unit to 1.

Change the y-axis scale so that it has a logarithmic scale, and a minimum of 1 x 106. Ignore the error message. This should enable you to “see” the other two plots.

Add titles to the graph and the two axes (including the units for the axes).

Choose:

Layout>Chart Title>Above Chart

Layout>Axis Title>Primary Horizontal Axis>Below Chart, and

Layout>Axis Title>Primary Vertical Axis>Rotated Title

Select each title and enter the appropriate text for each. See example below:

The legend was edited by right-clicking on the graph, and choosing Select Data, and editing each Series Name, changing each from the cell reference to simple text.

Yeah! You’re done with the graph.

How does the total amount of energy (area under the energy intensity curve) and wavelength of the maximum energy intensity change with increasing temperatures?

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